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<h1 class="title"><a class="reference external" href="http://www.osl.iu.edu/research/pbgl"><img align="middle" alt="Parallel BGL" class="align-middle" src="pbgl-logo.png" /></a> Betweenness Centrality</h1>

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<pre class="literal-block">
// named parameter versions
template&lt;typename Graph, typename Param, typename Tag, typename Rest&gt;
void
brandes_betweenness_centrality(const Graph&amp; g,
                               const bgl_named_params&lt;Param,Tag,Rest&gt;&amp; params);

template&lt;typename Graph, typename CentralityMap&gt;
void
brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality);

template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap&gt;
void
brandes_betweenness_centrality(const Graph&amp; g, CentralityMap centrality,
                               EdgeCentralityMap edge_centrality_map);

// non-named parameter versions
template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
         typename IncomingMap, typename DistanceMap, typename DependencyMap,
         typename PathCountMap, typename VertexIndexMap, typename Buffer&gt;
void
brandes_betweenness_centrality(const Graph&amp; g,
                               CentralityMap centrality,
                               EdgeCentralityMap edge_centrality_map,
                               IncomingMap incoming,
                               DistanceMap distance,
                               DependencyMap dependency,
                               PathCountMap path_count,
                               VertexIndexMap vertex_index,
                               Buffer sources,
                               typename property_traits&lt;DistanceMap&gt;::value_type delta);

template&lt;typename Graph, typename CentralityMap, typename EdgeCentralityMap,
         typename IncomingMap, typename DistanceMap, typename DependencyMap,
         typename PathCountMap, typename VertexIndexMap, typename WeightMap,
         typename Buffer&gt;
void
brandes_betweenness_centrality(const Graph&amp; g,
                               CentralityMap centrality,
                               EdgeCentralityMap edge_centrality_map,
                               IncomingMap incoming,
                               DistanceMap distance,
                               DependencyMap dependency,
                               PathCountMap path_count,
                               VertexIndexMap vertex_index,
                               Buffer sources,
                               typename property_traits&lt;WeightMap&gt;::value_type delta,
                               WeightMap weight_map);

// helper functions
template&lt;typename Graph, typename CentralityMap&gt;
typename property_traits&lt;CentralityMap&gt;::value_type
central_point_dominance(const Graph&amp; g, CentralityMap centrality);
</pre>
<p>The <tt class="docutils literal"><span class="pre">brandes_betweenness_centrality()</span></tt> function computes the
betweenness centrality of the vertices and edges in a graph.  The
method of calculating betweenness centrality in <em>O(V)</em> space is due to
Brandes <a class="citation-reference" href="#brandes01" id="id1">[Brandes01]</a>.  The algorithm itself is a modification of
Brandes algorithm by Edmonds <a class="citation-reference" href="#edmonds09" id="id2">[Edmonds09]</a>.</p>
<div class="contents topic" id="contents">
<p class="topic-title first">Contents</p>
<ul class="simple">
<li><a class="reference internal" href="#where-defined" id="id3">Where Defined</a></li>
<li><a class="reference internal" href="#parameters" id="id4">Parameters</a></li>
<li><a class="reference internal" href="#complexity" id="id5">Complexity</a></li>
<li><a class="reference internal" href="#algorithm-description" id="id6">Algorithm Description</a></li>
<li><a class="reference internal" href="#bibliography" id="id7">Bibliography</a></li>
</ul>
</div>
<div class="section" id="where-defined">
<h1><a class="toc-backref" href="#id3">Where Defined</a></h1>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/distributed/betweenness_centrality.hpp</span></tt>&gt;</p>
<p>also accessible from</p>
<p>&lt;<tt class="docutils literal"><span class="pre">boost/graph/betweenness_centrality.hpp</span></tt>&gt;</p>
</div>
<div class="section" id="parameters">
<h1><a class="toc-backref" href="#id4">Parameters</a></h1>
<dl class="docutils">
<dt>IN:  <tt class="docutils literal"><span class="pre">const</span> <span class="pre">Graph&amp;</span> <span class="pre">g</span></tt></dt>
<dd>The graph type must be a model of <a class="reference external" href="DistributedGraph.html">Distributed Graph</a>.  The graph
type must also model the <a class="reference external" href="http://www.boost.org/libs/graph/doc/IncidenceGraph.html">Incidence Graph</a> concept.  0-weighted
edges in <tt class="docutils literal"><span class="pre">g</span></tt> will result in undefined behavior.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">CentralityMap</span> <span class="pre">centrality</span></tt></dt>
<dd><p class="first">A centrality map may be supplied to the algorithm, if not supplied a
<tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no vertex centrality
information will be recorded.  The <tt class="docutils literal"><span class="pre">CentralityMap</span></tt> type must be a
<a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  The key type must be the graph's
vertex descriptor type.</p>
<p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span> <span class="pre">edge_centrality_map</span></tt></dt>
<dd><p class="first">An edge centrality map may be supplied to the algorithm, if not
supplied a <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt> will be used and no edge
centrality information will be recorded.  The <tt class="docutils literal"><span class="pre">EdgeCentralityMap</span></tt>
type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  The key type must be
the graph's vertex descriptor type.</p>
<p class="last"><strong>Default</strong>: A <tt class="docutils literal"><span class="pre">dummy_property_map</span></tt>.</p>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">IncomingMap</span> <span class="pre">incoming</span></tt></dt>
<dd><p class="first">The incoming map contains the incoming edges to a vertex that are
part of shortest paths to that vertex.  The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> type
must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  Its key type and value type
must both be the graph's vertex descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of <tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's vertex
descriptor type.</dd>
</dl>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">DistanceMap</span> <span class="pre">distance</span></tt></dt>
<dd><p class="first">The distance map records the distance to vertices during the
shortest paths portion of the algorithm.  The <tt class="docutils literal"><span class="pre">DistanceMap</span></tt> type
must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  Its key type must be the
graph's vertex descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
</dl>
</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">DependencyMap</span> <span class="pre">dependency</span></tt></dt>
<dd><p class="first">The dependency map records the dependency of each vertex during the
centrality calculation portion of the algorithm.  The
<tt class="docutils literal"><span class="pre">DependencyMap</span></tt> type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.  Its
key type must be the graph's vertex descriptor type.</p>
<dl class="last docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the value type of the <tt class="docutils literal"><span class="pre">CentralityMap</span></tt>.</dd>
</dl>
</dd>
</dl>
<p>IN:  <tt class="docutils literal"><span class="pre">PathCountMap</span> <span class="pre">path_count</span></tt></p>
<blockquote>
<p>The path count map records the number of shortest paths each vertex
is on during the centrality calculation portion of the algorithm.
The <tt class="docutils literal"><span class="pre">PathCountMap</span></tt> type must be a <a class="reference external" href="distributed_property_map.html">Distributed Property Map</a>.
Its key type must be the graph's vertex descriptor type.</p>
<dl class="docutils">
<dt><strong>Default</strong>: An <tt class="docutils literal"><span class="pre">iterator_property_map</span></tt> created from a</dt>
<dd><tt class="docutils literal"><span class="pre">std::vector</span></tt> of the graph's degree size type.</dd>
</dl>
</blockquote>
<dl class="docutils">
<dt>IN:  <tt class="docutils literal"><span class="pre">VertexIndexMap</span> <span class="pre">vertex_index</span></tt></dt>
<dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the vertex
descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt> and whose value type is an
integral type. The property map should map from vertices to their
(local) indices in the range <em>[0, num_vertices(g))</em>.</p>
<p class="last"><strong>Default</strong>: <tt class="docutils literal"><span class="pre">get(vertex_index,</span> <span class="pre">g)</span></tt></p>
</dd>
<dt>IN:  <tt class="docutils literal"><span class="pre">WeightMap</span> <span class="pre">weight_map</span></tt></dt>
<dd>A model of <a class="reference external" href="http://www.boost.org/libs/property_map/ReadablePropertyMap.html">Readable Property Map</a> whose key type is the edge
descriptor type of the graph <tt class="docutils literal"><span class="pre">g</span></tt>.  If not supplied the betweenness
centrality calculation will be unweighted.</dd>
<dt>IN: <tt class="docutils literal"><span class="pre">Buffer</span> <span class="pre">sources</span></tt></dt>
<dd><p class="first">A model of <a class="reference external" href="http://www.boost.org/libs/graph/doc/Buffer.html">Buffer</a> containing the starting vertices for the
algorithm.  If <tt class="docutils literal"><span class="pre">sources</span></tt> is empty a complete betweenness
centrality calculation using all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> will be
performed.  The value type of the Buffer must be the graph's vertex
descriptor type.</p>
<p class="last"><strong>Default</strong>: An empty <tt class="docutils literal"><span class="pre">boost::queue</span></tt> of int.</p>
</dd>
</dl>
</div>
<div class="section" id="complexity">
<h1><a class="toc-backref" href="#id5">Complexity</a></h1>
<p>Computing the shortest paths, counting them, and computing the
contribution to the centrality map is <em>O(V log V)</em>.  Calculating exact
betweenness centrality requires counting the shortest paths from all
vertices in <tt class="docutils literal"><span class="pre">g</span></tt>, thus exact betweenness centrality is <em>O(V^2 log
V)</em>.</p>
</div>
<div class="section" id="algorithm-description">
<h1><a class="toc-backref" href="#id6">Algorithm Description</a></h1>
<p>For the vertices in <tt class="docutils literal"><span class="pre">sources</span></tt> (or all vertices in <tt class="docutils literal"><span class="pre">g</span></tt> when
<tt class="docutils literal"><span class="pre">sources</span></tt> is empty) the algorithm first calls a customized
implementation of <a class="reference external" href="dijkstra_shortest_paths.html">delta_stepping_shortest_paths</a> which maintains a
shortest path tree using an <tt class="docutils literal"><span class="pre">IncomingMap</span></tt>.  The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt>
contains the source of all incoming edges on shortest paths.</p>
<p>The <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> defines the shortest path DAG at the target of the
edges in the shortest paths tree.  In the bidirectional case edge
flags could be used to translate the shortest paths information to the
source of the edges.  Setting edge flags during the shortest path
computation rather than using an <tt class="docutils literal"><span class="pre">IncomingMap</span></tt> would result in
adding an <em>O(V)</em> factor to the inner loop of the shortest paths
computation to account for having to clear edge flags when a new
shortest path is found.  This would increase the complexity of the
algorithm.  Asymptotically, the current implementation is better,
however using edge flags in the bidirectional case would reduce the
number of supersteps required by the depth of the shortest paths DAG
for each vertex.  Currently an <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is explicitly
constructed by simply reversing the edges in the incoming map.  Once
the <tt class="docutils literal"><span class="pre">outgoing</span></tt> map is constructed it is traversed in dependency
order from the source of the shortest paths calculation in order to
compute path counts.  Once path counts are computed the shortest paths
DAG is again traversed in dependency order from the source to
calculate the dependency and centrality of each vertex.</p>
<p>The algorithm is complete when the centrality has been computed from
all vertices in <tt class="docutils literal"><span class="pre">g</span></tt>.</p>
</div>
<div class="section" id="bibliography">
<h1><a class="toc-backref" href="#id7">Bibliography</a></h1>
<table class="docutils citation" frame="void" id="brandes01" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id1">[Brandes01]</a></td><td>Ulrik Brandes.  A Faster Algorithm for Betweenness
Centrality. In the Journal of Mathematical Sociology, volume 25
number 2, pages 163--177, 2001.</td></tr>
</tbody>
</table>
<table class="docutils citation" frame="void" id="edmonds09" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
<tr><td class="label"><a class="fn-backref" href="#id2">[Edmonds09]</a></td><td>Nick Edmonds, Torsten Hoefler, and Andrew Lumsdaine.
A Space-Efficient Parallel Algorithm for Computing Betweenness
Centrality in Sparse Networks.  Indiana University tech report.
2009.</td></tr>
</tbody>
</table>
<hr class="docutils" />
<p>Copyright (C) 2009 The Trustees of Indiana University.</p>
<p>Authors: Nick Edmonds and Andrew Lumsdaine</p>
</div>
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